Termination w.r.t. Q proof of /tmp/tmp3KXokr/Ex14_Luc06

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

↳5 QDPOrderProof (⇔)

↳6 QDP

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__H(X) → A__G(mark(X), X)
A__H(X) → MARK(X)
A__G(a, X) → A__F(b, X)
A__F(X, X) → A__H(a__a)
A__F(X, X) → A__A
MARK(h(X)) → A__H(mark(X))
MARK(h(X)) → MARK(X)
MARK(g(X1, X2)) → A__G(mark(X1), X2)
MARK(g(X1, X2)) → MARK(X1)
MARK(a) → A__A
MARK(f(X1, X2)) → A__F(mark(X1), X2)
MARK(f(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__G(a, X) → A__F(b, X)
A__F(X, X) → A__H(a__a)
A__H(X) → A__G(mark(X), X)
A__H(X) → MARK(X)
MARK(h(X)) → A__H(mark(X))
MARK(h(X)) → MARK(X)
MARK(g(X1, X2)) → A__G(mark(X1), X2)
MARK(g(X1, X2)) → MARK(X1)
MARK(f(X1, X2)) → A__F(mark(X1), X2)
MARK(f(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__H(X) → MARK(X)
MARK(h(X)) → A__H(mark(X))
MARK(h(X)) → MARK(X)
MARK(g(X1, X2)) → A__G(mark(X1), X2)
MARK(g(X1, X2)) → MARK(X1)
MARK(f(X1, X2)) → A__F(mark(X1), X2)
MARK(f(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__G(x1, x2) = A__G(x1)
a = a
A__F(x1, x2) = A__F(x1)
b = b
A__H(x1) = A__H(x1)
a__a = a__a
mark(x1) = x1
MARK(x1) = MARK(x1)
h(x1) = h(x1)
g(x1, x2) = g(x1)
f(x1, x2) = f(x1)
a__h(x1) = a__h(x1)
a__g(x1, x2) = a__g(x1)
a__f(x1, x2) = a__f(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[h₁, g₁, f₁, a_{_h₁}, a_{_g₁}, a_{_f₁}] > [A_{_G₁}, A_{_F₁}, A_{_H₁}] > [a, b, a_{_a}, MARK₁]

Status:

A_{_G₁}: multiset
a: multiset
A_{_F₁}: multiset
b: multiset
A_{_H₁}: multiset
a_{_a}: multiset
MARK₁: multiset
h₁: multiset
g₁: multiset
f₁: multiset
a_{_h₁}: multiset
a_{_g₁}: multiset
a_{_f₁}: multiset

The following usable rules [FROCOS05] were oriented:

a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

(6) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__G(a, X) → A__F(b, X)
A__F(X, X) → A__H(a__a)
A__H(X) → A__G(mark(X), X)

The TRS R consists of the following rules:

a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.